In Chapter 7 we have studied the construction of 4d N =1 heterotic string vacua by compactification on smooth CY manifolds. The analysis of such compactifications is however limited because their worldsheet theory is not exactly solvable. Therefore, we must rely on the Kaluza–Klein reduction of the 10d supergravity action, which provides a good approximation to the 4d physics only in the large volume regime. In this chapter we study 4d N =1 heterotic string vacua obtained from toroidal orbifold compactifications and other exact CFT constructions, which overcome this limitation. These are simple α′-exact compactifications, which share many properties with more general CY compactifications (and in fact are often closely related to them), and which allow a very explicit construction of phenomenologically interesting particle physics models. We describe toroidal orbifolds, which provide a free CFT description of geometries which can be regarded as singular limits of CY spaces. We also introduce asymmetric orbifolds and free fermionic models, which are also described by free worldsheet CFTs but in general do not admit a geometric interpretation. Finally we describe Gepner models (and orbifolds thereof), defined by interacting but solvable CFTs and which can often be regarded as compactifications on CY spaces of stringy size. We focus on the E8 × E8 heterotic string theory, although the basic rules apply in complete analogy to the SO(32) theory.

Toroidal orbifolds

Let us start by describing the geometry of toroidal orbifolds. The main ingredients are useful for heterotic string compactifications, as well as for other string theories, like type II orientifolds.